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A finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2).

Abstract

Let Bq be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix q L [k.sup.[theta]x[theta]]. Let [L.sub.q] be the Lusztig algebra associated to [B.sub.q] [AAR]. We present [L.sub.q] as an extension (as braided Hopf algebras) of [B.sub.q] by Zq where Zq is isomorphic to the universal enveloping algebra of a Lie algebra [n.sub.q]. We compute the Lie algebra [n.sub.q] when [theta] = 2.

1 Introduction

1.1 Let k be a field, algebraically closed and of characteristic zero. Let [theta] [member of] N, I = [I.sub.[theta]] := {1,2, ... , [theta]}. Let q = [([q.sub.ij])i,.sub.j[member of]I] be a matrix with entries in [k.sup.x], V a vector space with a basis ([[x.sub.i]).sub.i[member of]I] and [c.sup.q] [member of] GL(V [cross product] V) be given by

[c.sup.q]([x.sub.i] [cross product] [x.sub.j]) = [q.sub.ij][x.sub.j] [cross product] [x.sub.i], i,j [cross product] I.

Then ([c.sup.q] [cross product] id)(id [cross product] [c.sup.q])([c.sup.q] [cross product] id) = (id[cross product][c.sup.q])([c.sup.q] [cross product] id)(id[cross product][c.sup.q]), i.e. (V,[c.sup.q]) is a braided vector space and the corresponding Nichols algebra [B.sub.q] := B(V) is called of diagonal type. Recall that [B.sub.q] is the image of the unique map of braided Hopf algebras [OHM] : T(V) [right arrow] [T.sub.c](V) from the free associative algebra of V to the free associative coalgebra of V, such that [[OHM].sub.|V] = i[d.sub.[V.sub.. For unexplained terminology and notation, we refer to [AS].

Remarkably, the explicit classification of all q such that dim [B.sub.q] < [infinity] is known [H2] (we recall the list when [theta] = 2 in Table 1). Also, for every q in the list of [H2], the defining relations are described in [A2, A3].

1.2 Assume that dim [B.sub.q] < [infinity]. Two infinite dimensional graded braided Hopf algebras [B.sub.q] and [C.sub.q] (the Lusztig algebra of V) were introduced and studied in [A3, A5], respectively [AAR]. Indeed, [B.sub.q] is a pre-Nichols, and [L.sub.q] a post-Nichols, algebra of V, meaning that [B.sub.q] is intermediate between T(V) and [B.sub.q], while [L.sub.q] is intermediate between [B.sub.q] and [T.sub.c](V). This is summarized in the following commutative diagram:

[ILLUSTRATION OMITTED]

The algebras [B.sub.q] and [L.sub.q] are generalizations of the positive parts of the De Concini-Kac-Procesi quantum group, respectively the Lusztig quantum divided powers algebra. The distinguished pre-Nichols algebra [B.sub.q] is defined discarding some of the relations in [A3], while [L.sub.q] is the graded dual of [B.sub.q].

1.3 The following notions are discussed in Section 2. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the generalized positive root system of [B.sub.q] and let [D.sub.q] [subset] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the set of Cartan roots of q. Let x[beta] be the root vector associated to [beta] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], let [N.sub.[beta]] = ord [q.sub.[beta][beta]] and let [Z.sub.q] be the subalgebra of [B.sub.q] generated by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [beta] [member of] [D.sub.q]. By [A5, Theorems 4.10, 4.13], [Z.sub.q] is a braided normal Hopf subalgebra of [B.sub.q] and [Z.sub.q] = [.sup.co[pi]][B.sub.q]. Actually, [Z.sub.q] is a true commutative Hopf algebra provided that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Let Zq be the graded dual of [Z.sub.q]; under the assumption (1) Zq is a cocommutative Hopf algebra, hence it is isomorphic to the enveloping algebra u([n.sub.q]) of the Lie algebra [n.sub.q] := P(Zq). We show in Section 3 that [C.sub.q] is an extension (as braided Hopf algebras) of [B.sub.q] by Zq:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

The main result of this paper is the determination of the Lie algebra [n.sub.q] when [theta] = 2 and the generalized Dynkin diagram of q is connected.

Theorem 1.1. Assume that dim [B.sub.q] < [infinity] and [theta] = 2. Then [n.sub.q] is either 0 or isomorphic to [g.sub.+], where g is a finite-dimensional semisimple Lie algebra listed in the last column of Table 1.

Assume that there exists a Cartan matrix a = ([a.sub.ij]) of finite type, that becomes symmetric after multiplying with a diagonal ([d.sub.i]), and a root of unit q of odd order (and relatively prime to 3 if a is of type [G.sub.2]) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i, j [member of] I. Then (2) encodes the quantum Frobenius homomorphism defined by Lusztig and Theorem 1.1 is a result from [L].

The penultimate column of Table 1 indicates the type of q as established in [AA]. Thus, we associate Lie algebras in characteristic zero to some contragredient Lie (super)algebras in positive characteristic. In a forthcoming paper we shall compute the Lie algebra [n.sub.q] for [theta] > 2.

1.4 The paper is organized as follows. We collect the needed preliminary material in Section 2. Section 3 is devoted to the exactness of (2). The computations of the various [n.sub.q] is the matter of Section 4. We denote by GN the group of N-th roots of 1, and by [G'.sub.N] its subset of primitive roots.

2 Preliminaries

2.1 The Nichols algebra, the distinguished-pre-Nichols algebra and the Lusztig algebra

Let q be as in the Introduction and let (V, [c.sup.q]) be the corresponding braided vector space of diagonal type. We assume from now on that [B.sub.q] is finite-dimensional. Let [([[alpha].sub.j]).sub.j[member of]I] be the canonical basis of [Z.sup.[theta]]. Let q : [Z.sup.[theta]] x [Z.sup.[theta]] [right arrow]* [k.sup.x] be the Z-bilinear form associated to the matrix q, i.e. q([[alpha].sub.j], [[alpha].sub.k]) = [q.sub.jk] for all j,k [member of] I. If [alpha],[beta] [member of] [Z.sup.[theta]], we set [q.sub.[alpha][beta]] = q([alpha], [beta]). Consider the matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

This is well-defined by [R]. Let i [member of] I. We recall the following definitions:

* The reflection [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* The matrix [[rho].sub.i](q), given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* The braided vector space [[rho].sub.i](V) of diagonal type with matrix [[rho].sub.i](q).

A basic result is that [B.sub.q] [equivalent] B [[rho].sub.i](q),at leastas graded vector spaces.

The algebras T(V) and [B.sub.q] are [Z.sup.[theta]]-graded by degxi = [[alpha].sub.i], i [member of] I. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the set of [Z.sup.[theta]]-degrees of the generators of a PBW-basis of [B.sub.q], counted with multiplicities [H1]. The elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are called (positive) roots. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let

X := {[rho][j.sub.1] ... [rho][j.sub.[N.sub.(q) : [j.sub.1], ... ,jN [member of] I,N [member of] N}.

Then the generalized root system of q is the fibration [DELTA] [right arrow]* X, where the fiber of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The Weyl groupoid of [B.sub.q] is a groupoid, denoted [W.sub.q],

that acts on this fibration, generalizing the classical Weyl group, see [H1]. We know from loc. cit. that [W.sub.q] is finite (and this characterizes finite-dimensional Nichols algebras of diagonal type).

Here is a useful description of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let w [member of] [W.sub.q] be an element of maximal length. We fix a reduced expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For 1 [less than or equal to] k [less than or equal to] M set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [CH, Prop. 2.12]; in particular [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The notion of Cartan root is instrumental for the definitions of [B.sub.q] and [L.sub.q]. First, following [A5] we say that i [member of] I is a Cartan vertex of q if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

Then the set of Cartan roots of q is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given a positive root [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there is an associated root vector [x.sub.[beta]] [member of] [B.sub.q] defined via the so-called Lusztig isomorphisms [H3]. Set [N.sub.[beta]] = ord [q.sub.[beta][beta]] [member of] N, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For simplicity, we introduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all k [member of] IM}. (6)

By [A5, Theorem 3.6] the set {[x.sup.h] | h [member of] H} is a basis of [B.sub.q].

As said in the Introduction, the Lusztig algebra associated to [B.sub.q] is the braided Hopf algebra Lq which is the graded dual of [B.sub.q]. Thus, it comes equipped with a bilinear form <,> : [B.sub.q] x [L.sub.q] [right arrow] k, which satisfies for all x,x' [member of] [B.sub.q], y, y' [member of] [L.sub.q]

<y,xx'> = <[y.sup.(2)], x><[y.sup.(1)],x'> and <yy',x> = <y,[x.sup.(2)]><y' ,[x.sup.(1)]>.

If h [member of] H, then define [y.sub.h] [member of] [L.sub.q] by ([y.sub.h],[x.sup.j]) = [[delta].sub.h,j], j [member of] H. Let [([h.sub.k]).sub.k[member of]IM] denote the canonical basis of [Z.sup.M]. If k [member of] IM and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we denote the element [y.sub.n][h.sub.k] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the algebra [L.sub.q] is generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

by [AAR]. Moreover, by [AAR, 4.6], the following set is a basis of [L.sub.q]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2.2 Lyndon words, convex order and PBW-basis

For the computations in Section 4 we need some preliminaries on Kharchenko's PBW-basis. Let (V, q) be as above and let X be the set of words with letters in X = {[x.sub.1], ... , [x.sub.[theta]]} (our fixed basis of V); the empty word is 1 and for u [member of] X we write l(u) the length of u. We can identify kX with T(V).

Definition 2.1. Consider the lexicographic order in X. We say that u [member of] X - {1} is a Lyndon word if for every decomposition u = vw, v, w [member of] X - {1}, then u < w. We denote by L the set of all Lyndon words.

A well-known theorem, due to Lyndon, established that any word u [member of] X admits a unique decomposition, named Lyndon decomposition, as a non-increasing product of Lyndon words:

u = [l.sub.1][l.sub.2] ... [l.sub.r], [l.sub.i] [member of] L, [l.sub.r] [less than or equal to] ... [less than or equal to] [l.sub.1]. (7)

Also, each [l.sub.i] [member of] L in (7) is called a Lyndon letter of u.

Now each u [member of] L - X admits at least one decomposition u = [v.sub.1][v.sub.2] with [v.sub.1],[v.sub.2] [member of] L. Then the Shirshov decomposition of u is the decomposition u = [u.sub.1] [u.sub.2], [u.sub.1], [u.sub.2] [member of] L, such that [u.sub.2] is the smallest end of u between all possible decompositions of this form.

For any braided vector space V, the braided bracket of x,y [member of] T(V) is

[[x,y].sub.c] := multiplication * (id -c) (x [cross product] y). (8)

Using the identification T(V) = kX and the decompositions described above, we can define a k-linear endomorphism [[-].sub.c] of T(V) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

We will describe PBW-bases using this endomorphism.

Definition 2.2. For l [member of] L, the element [[l].sub.c] is the corresponding hyperletter. A word written in hyperletters is an hyperword; a monotone hyperword is an hyperword [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [u.sub.1] > *** > [u.sub.m].

Consider now a different order on X, called deg-lex order [K]: For each pair u,v [member of] X, we have that u [??] v if l(u) < l(v), or l(u) = l(v) and u > v for the lexicographical order. This order is total, the empty word 1 is the maximal element and it is invariant by left and right multiplication.

Let I be a Hopf ideal of T(V) and R = T(V)/I. Let [pi] : T(V) [right arrow] R be the canonical projection. We set:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, if u [member of] [G.sub.[I.sub. and u = vw, then v,w [member of] [G.sub.[I.sub.. So, each u [member of] [G.sub.[I.sub. is a non-increasing product of Lyndon words of [G.sub.[I.sub..

Let [S.sub.[I.sub.:= [G.sub.[I.sub. [intersection] L and let [h.sub.[I.sub. : [S.sub.[I.sub. [right arrow] {2,3, ... } [union] {[infinity]} be defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Theorem 2.3. [K] The following set is a PBW-basis ofR = T(V)/I:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. []

We refer to this base as Kharchenko's PBW-basis of T(V)/I (it depends on the order of X).

Definition 2.4. [A2, 2.6] Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be as above and let < be a total order on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We say that the order is convex if for each [alpha], [beta] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [alpha] < [beta] and [alpha] + [beta] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [alpha] < [alpha] + [beta] < [beta]. The order is called strongly convex if for each ordered subset [[alpha].sub.1] [less than or equal to] [[alpha].sub.2] [less than or equal to] ... [less than or equal to] [[alpha].sub.k] of elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [alpha] = [[summation].sub.i] [[alpha].sub.i] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then [[alpha].sub.1] < [alpha] < [[alpha].sub.k].

Theorem 2.5. [A2, 2.11] The following statements are equivalent:

* The order is convex.

* The order is strongly convex.

* The order arises from a reduced expression of a longest element w [member of] [W.sub.q], cf. (4). []

Now, we have two PBW-basis of [B.sub.q] (and correspondingly of [B.sub.q]), namely Kharchenko's PBW-basis and the PBW-basis defined from a reduced expression of a longest element of the Weyl groupoid. But both basis are reconciled by [AY, Theorem 4.12], thanks to [A2, 2.14]. Indeed, each generator of Kharchenko's PBW-basis is a multiple scalar of a generator of the secondly mentioned PBW-basis. So, for ease of calculations, in the rest of this work we shall use the Kharchenko generators.

The following proposition is used to compute the hyperword [[[l.sub.[beta]]].sub.c] associated to a root [beta] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

Proposition 2.6. [A2, 2.17] For [beta] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] []

We give a list of the hyperwords appearing in the next section:
               Root                           Hyperword

          [[alpha].sub.i]                     [x.sub.i]
n[[alpha].sub.1] + [[alpha].sub.2]           [(a[d.sub.c]
                                      [x.sub.1]).sup.n][x.sub.2]
[[alpha].sub.1] + 2[[alpha].sub.2]   [MATHEMATICAL EXPRESSION NOT
                                        REPRODUCIBLE IN ASCII]
3[[alpha].sub.1] + 2[[alpha].sub.2]  [MATHEMATICAL EXPRESSION NOT
                                        REPRODUCIBLE IN ASCII]
4[[alpha].sub.1] + 3[[alpha].sub.2]  [MATHEMATICAL EXPRESSION NOT
                                        REPRODUCIBLE IN ASCII]

5[[alpha].sub.1] + 3[[alpha].sub.2]  [MATHEMATICAL EXPRESSION NOT
                                        REPRODUCIBLE IN ASCII]

               Root                               Notation

          [[alpha].sub.i]                          [x.sub.i]
n[[alpha].sub.1] + [[alpha].sub.2]
                                               [x.sub.1 ... 12]
[[alpha].sub.1] + 2[[alpha].sub.2]
                                       [[[x.sub.12],[x.sub.2]].sub.c]
3[[alpha].sub.1] + 2[[alpha].sub.2]
                                      [[[x.sub.112],[x.sub.12]].sub.c]
4[[alpha].sub.1] + 3[[alpha].sub.2]
                                     [[[[x.sub.112],[x.sub.12]].sub.c],
                                             [x.sub.12]].sub.c]
5[[alpha].sub.1] + 3[[alpha].sub.2]
                                       [[[[x.sub.112], [[x.sub.112],
                                         [x.sub.12]].sub.c]].sub.c]


We use an analogous notation for the elements of [L.sub.q]: for example we write y112,12 when we refer to the element of [L.sub.q] which corresponds to [[[x.sub.112], [x.sub.12]].sub.c].

3 Extensions of braided Hopf algebras

We recall the definition of braided Hopf algebra extensions given in [AN]; we refer to [BD, GG] for more general definitions. Below we denote by [DELTA] the coproduct of a braided Hopf algebra A and by [A.sup.+] the kernel of the counit.

First, if [pi] : C [right arrow] B is a morphism of Hopf algebras in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Definition 3.1. [AN, [section]2.5] Let H be a Hopf algebra. A sequence of morphisms of Hopf algebras in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

is an extension of braided Hopf algebras if

(i) i is injective,

(ii) [pi] is surjective,

(iii) ker [pi] = Ci([A.sup.+]) and

(iv) A = [C.sup.co[pi]], or equivalently A = [.sup.co[pi]]C.

For simplicity, we shall write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead of (10).

This Definition applies in our context: recall that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let Zq be the subalgebra of [B.sub.q] generated by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

* The inclusion i : [Z.sub.q] [right arrow]* [B.sub.q] is injective and the projection [pi] : [B.sub.q] [right arrow] [B.sub.q] is surjective.

* [A5, Theorem 4.10] [Z.sub.q] is a normal Hopf subalgebra of [B.sub.q]; since ker [pi] is the two-sided ideal generated by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

* [A5, Theorem 4.13] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence we have an extension of braided Hopf algebras

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

The morphisms i and [pi] are graded. Thus, taking graded duals, we obtain a new sequence of morphisms of braided Hopf algebras

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Proposition 3.2. The sequence (2) is an extension of braided Hopf algebras.

Proof. The argument of [A, 3.3.1] can be adapted to the present situation, or more generally to extensions of braided Hopf algebras that are graded with finite-dimensional homogeneous components. The map [pi]* : [B.sub.q] [right arrow] [L.sub.q] is injective because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is surjective being the transpose of a graded monomorphism between two locally finite graded vector spaces. Now, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Similarly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because ker [[pi].sup.[perpendicular to]] = [B.sub.q]. []

From now on, we assume the condition (1) on the matrix q mentioned in the Introduction, that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following result is our basic tool to compute the Lie algebra [n.sub.q].

Theorem 3.3. The braided Hopf algebra Zq is an usual Hopf algebra, isomorphic to the universal enveloping algebra of the Lie algebra [n.sub.q] = P(Zq). The elements [[xi].sub.[beta]] := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], form a basis of [n.sub.q].

Proof. Let [A.sub.q] be the subspace of [L.sub.q] generated by the ordered monomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are all the Cartan roots of [B.sub.q] and [r.sub.1], ... ,[r.sub.k] [member of] [N.sub.0]. We claim that the restriction of the multiplication [epsilon] : [B.sub.q] [cross product] [A.sub.q] [right arrow] [L.sub.q] is an isomorphism of vector spaces. Indeed, [epsilon] is surjective by the commuting relations in [L.sub.q]. Also, the Hilbert series of [L.sub.q], [B.sub.q] and [A.sub.q] are respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since the multiplication is graded and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is injective. The claim follows and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

We next claim that i* : [A.sub.q] [right arrow]* Zq is an isomorphism of vector spaces. Indeed, by (12), ker [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By (13), the claim follows.

By (1), [Z.sub.q] is a commutative Hopf algebra, see [A5]; hence [Z.sub.q] is a cocommutative Hopf algebra. Now the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are primitive, i.e. belong to [n.sub.q] = P([Z.sub.q]). The monomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [r.sub.1], ... ,[r.sub.k] [member of] [N.sub.0] form a basis of [Z.sub.q], hence

[Z.sub.q] = k([[xi].sub.[beta]] : [beta] [member of] [D.sub.q]) [[subset].bar] U([n.sub.q]) [[subset].bar] [Z.sub.q].

We conclude that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a basis of [n.sub.q] and that Zq = U([n.sub.q]). []

4 Proof of Theorem 1.1

In this section we consider all indecomposable matrices q of rank 2 whose associated Nichols algebra [B.sub.q] is finite-dimensional; these are classified in [H2] and we recall their diagrams in Table 1. For each q we obtain an isomorphism between Zq and U([g.sub.+]), the universal enveloping algebra of the positive part of g. Here g is the semisimple Lie algebra of the last column of Table 1, with Cartan matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By simplicity we denote g by its type, e.g. g = [A.sub.2].

We recall that we assume (1) and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus,

[[[xi].sub.[alpha]],[[[xi].sub.[beta]]].sub.c] = [[xi].sub.[alpha]][[xi].sub.[beta]] - [[xi].sub.[beta]][[xi].sub.[alpha]] = [[[xi].sub.[alpha]],[[xi].sub.[beta]]], for all [alpha] ,[beta] [member of] [D.sub.q].

The strategy to prove the isomorphism F : U([g.sub.+]) [right arrow]* [Z.sub.q] is the following:

1. If Dq = [empty set], then [g.sub.+] = 0. If |[D.sub.q]| = 1, then g = sl2, i.e. of type [A.sub.1].

2. If |Dq| = 2, then g is of type [A.sub.1] [direct sum] [A.sub.1]. Indeed, let [D.sub.q] = {[alpha], [beta]}. As [Z.sub.q] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-graded, [[[xi].sub.[alpha]], [[xi].sub.[beta]]] [member of] [n.sub.q] has degree [N.sub.[alpha]][alpha] + [N.sub.[beta]][beta]. Thus [[[xi].sub.[alpha]], [[xi].sub.[beta]]] = 0.

3. Now assume that |[D.sub.q]| > 2. We recall that [Z.sub.q] is generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a primitive element of [B.sub.q]}.

We compute the coproduct of all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [B.sub.q], [beta] [member of] [D.sub.q], using that [[DELTA].bar] is a graded map and [Z.sub.q] is a Hopf subalgebra of [B.sub.q]. In all cases we get two primitive elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], thus [Z.sub.q] is generated bv [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4. Using the coproduct again, we check that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

To prove (14), it is enough to observe that [n.sub.q] has a trivial component of degree [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now (14) implies that there exists a surjective map of Hopf algebras F : U([g.sub.+]) [??] Zq such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

5. To prove that F is an isomorphism, it suffices to see that the restriction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an isomorphism; but in each case we see that * is surjective, and dim [g.sub.+] = dim [n.sub.q] = |Dq|.

We refer to [A1, AAY, A4] for the presentation, root system and Cartan roots of braidings of standard, super and unidentified type respectively.

Row 1. Let q [member of] [G.sub.[N.sub., N [greater than or equal to] 2. The diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] corresponds to a braiding of Cartan type [A.sub.2] whose set of positive roots is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [N.sub.[beta]] = N for all [beta] [greater than or equal to] [D.sub.q]. By hypothesis, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The elements [x.sub.1], [x.sub.2] [greater than or equal to] [B.sub.q] are primitive and

[[DELTA].bar]([x.sub.12]) = [x.sub.12] [cross product] 1 + 1 [cross product] [x.sub.12] + (1 - [q.sup.-1])[x.sub.1] [cross product] [x.sub.2].

Then the coproducts of the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As [[[xi].sub.2], [[xi].sub.12]], [[[xi].sub.1], [[xi].sub.12]] [member of] [n.sub.q] have degree N[[alpha].sub.1] + 2N[[alpha].sub.2], respectively 2N[[alpha].sub.1] + N[[alpha].sub.2], and the components of these degrees of [n.sub.q] are trivial, we have

[[[xi].sub.2], [[xi].sub.12]] = [[[xi].sub.1], [[xi].sub.12]] = 0.

Again by degree considerations, there exists c [member of] k such that [[[xi].sub.2],[[xi].sub.1]] = c[[xi].sub.12]. By the duality between [Z.sub.q] and [Z.sub.q] we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then there exists a morphism of algebras [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by

[e.sub.1] [right arrow] [[xi].sub.1], [e.sub.2] [right arrow] [[xi].sub.2].

This morphism takes a basis of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to a basis of [n.sub.q], so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Row 2. Let q [member of] [G'.sub.N'], N [greater than or equal to] 3. These diagrams correspond to braidings of super type A with positive roots [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first diagram is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] In this case the unique Cartan root is [[alpha].sub.1] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is primitive and Zq is generated by [[xi].sub.1]. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second diagram gives a similar situation, since [D.sub.q] = {[[alpha].sub.1] + [[alpha].sub.2]}.

Row 3. Let q [member of] [G'.sub.N'], N [greater than or equal to] 3. The diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] corresponds to a braiding of Cartan type [B.sub.2] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The coproducts of the generators of [B.sub.q] are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have two different cases depending on the parity of N.

1. If N is odd, then [N.sub.[beta]] = N for all [beta] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for some C [member of] k. Hence, in [Z.sub.q] we have the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus there exists an algebra map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Moreover, F is an isomorphism, and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using the relations of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we check that C = 2[(1 - [q.sup.-1]).sup.N][(1 - [q.sup.-2]).sup.N].

(2) If N = 2M > 2, then [N.sub.[alpha]1] = [N.sub.[alpha]1+[alpha]2] = N and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, the following relations hold in [Z.sub.q]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is an isomorphism of algebras. (Of course [C.sub.2] [equivalent] [B.sub.2] but in higher rank we will get different root systems depending on the parity of N).

Row 4. Let q [member of] [G'.sub.N'], N [not equal to] 2,4. These diagrams correspond to braidings of super type B with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If the diagram is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the Cartan roots are [[alpha].sub.1] and [[alpha].sub.1] + [[alpha].sub.2], with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; here, M = N if N is odd and M = [[N]/2] if N is even. The elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are primitive in [B.sub.q]. Thus, in [Z.sub.q], [[[xi].sub.12],[[xi].sub.1]] = 0 and [Z.sub.q] [equivalent] U[(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

If we consider the diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then [D.sub.q] = {[[alpha].sub.1],[[alpha].sub.1] + [[alpha].sub.2]}, [N.sub.[alpha]1] = M and [N.sub.[alpha]1+[alpha]2] = N. The elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are primitive, so [[[xi].sub.12],[[xi].sub.1]] = 0 and [Z.sub.q] [equivalent] U [(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

Row 5. Let q [member of] [G'.sub.N'], N [not equal to] 3, [zeta] [member of] [G'.sub.3]. The diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] corresponds to a braiding of standard type [B.sub.2], so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The other diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is obtained by changing the parameter q [??]* [zeta][q.sup.-1].

The Cartan roots are 2[[alpha].sub.1] + [[alpha].sub.2] and [[alpha].sub.2], with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [N.sub.[alpha]2] = N. The elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are primitive. Thus, in [Z.sub.q], we have [[[xi].sub.112], [[xi].sub.2]] = 0. Hence, [Z.sub.q] [equivalent] [u(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

Row 6. Let [zeta] [member of] G'3. The diagrams [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] correspond to braidings of standard type B, thus @#@. In both cases [D.sub.q] is empty so the corresponding Lie algebras are trivial.

Row 7. Let [zeta] [member of] [G'.sub.12]. The diagrams of this row correspond to braidings of type ufo(7). In all cases [D.sub.q] = [empty set] and the associated Lie algebras are trivial.

Row 8. Let [zeta] [member of] [G'.sub.12]. The diagrams of this row correspond to braidings of type ufo(8). For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case [D.sub.q] = {[[alpha].sub.1] + [[alpha].sub.2]}, [N.sub.[alpha]1+[alpha]2] = 12. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The same result holds for the other braidings in this row.

Row 9. Let [zeta] [member of] [G'.sub.9]. The diagrams of this row correspond to braidings of type brj(2; 3). If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

In this case [D.sub.q] = {[[alpha].sub.1],[[alpha].sub.1] + [[alpha].sub.2]} and [N.sub.[alpha]1] = [N.sub.[alpha]1+[alpha]2] = 18. Thus [[[xi].sub.12],[[xi].sub.1]] = 0, so [Z.sub.q] [equivalent] [U(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the set of positive roots are, respectively,

{[[alpha].sub.1],2[[alpha].sub.1] + [[alpha].sub.2],3[[alpha].sub.1] + 2[[alpha].sub.2],4[[alpha].sub.1] + 3[[alpha].sub.2],[[alpha].sub.1] + [[alpha].sub.2], [[alpha].sub.2]}, {[[alpha].sub.1],4[[alpha].sub.1] + [[alpha].sub.2],3[[alpha].sub.1] + [[alpha].sub.2],2[[alpha].sub.1] + [[alpha].sub.2],[[alpha].sub.1] + [[alpha].sub.2], [[alpha].sub.2]},

the Cartan roots are, respectively, [[alpha].sub.1] + [[alpha].sub.2],2[[alpha].sub.1] + [[alpha].sub.2] and [[alpha].sub.1],2[[alpha].sub.1] + [[alpha].sub.2]. Hence, in both cases, [Z.sub.q] [equivalent] [U(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

Row 10. Let q [member of] [G'.sub.[N.sub., N [greater than or equal to] 4. The diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] corresponds to a braiding of Cartan type [G.sub.2], so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The coproducts of the PBW-generators are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have two cases.

1. If 3 does not divide N, then [N.sub.[beta]] = N for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, in [B.sub.q],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

for some [a.sub.i] [member of] k. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not primitive. Hence [Z.sub.q] is generated by [[xi].sub.1] and [[xi].sub.2]; also

[[[xi].sub.2], [[xi].sub.1]] = [a.sub.1] [[xi].sub.12]; [[[xi].sub.12],[[xi].sub.1]] = [a.sub.2] [[xi].sub.112]; [[[xi].sub.112], [[xi].sub.1]] = [a.sub.4] [[xi].sub.1112]; [[[xi].sub.1], [[xi].sub.1112]] = [[[xi].sub.2],[[xi].sub.12]] = 0.

Thus, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) If N = 3M, then [N.sub.[alpha]1] = [N.sub.[alpha]1+[alpha]2] = [N.sub.2[alpha]1+[alpha]2] = N and [N.sub.3[alpha]1+[alpha]2] = [N.sub.3[alpha]1+2[alpha]2] = [N.sub.[alpha]2] = M. In this case we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some [b.sub.i] [member of] k. We compute some of them explicitly:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As these scalars are not zero, the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not primitive. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Row 11. Let [zeta] [member of] [G'.sub.8]. The diagrams of this row correspond to braidings of standard type [G.sub.2], so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the Cartan roots are 2[[alpha].sub.1] + [[alpha].sub.2] and [[alpha].sub.2] with [N.sub.2[alpha]1+[alpha]2] = [N.sub.[alpha]2] = 8. The elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are primitive and [[[xi].sub.2], [[xi].sub.112]] = 0 in [Z.sub.q]. Hence [Z.sub.q] [equivalent] [u(([A.sub.1] [direct sum] [A.sub.1]).sup.+]). An analogous result holds for the other diagrams of the row.

Row 12. Let [zeta] [member of] [G'.sub.24]. This row corresponds to type ufo (9). If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [D.sub.q] = {[[alpha].sub.1] + [[alpha].sub.2],3[[alpha].sub.1] + [[alpha].sub.2]}. Here, [N.sub.[alpha]1+[alpha]2] = [N3.sub.[alpha]1+[alpha]2] = 24, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are primitive. In [Z.sub.q] we have the relation [[[xi].sub.12],[[xi].sub.1112]] = 0; thus [Z.sub.q] [equivalent] [u(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

F or the other diagrams, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the sets of positive roots are, respectively,

{[[alpha].sub.1], [[alpha].sub.1] + [[alpha].sub.2], 2[[alpha].sub.1] + [[alpha].sub.2], 3[[alpha].sub.1] + [[alpha].sub.2],3[[alpha].sub.1] + 2[[alpha].sub.2], 5[[alpha].sub.1] + 2[[alpha].sub.2], 5[[alpha].sub.1] + 3[[alpha].sub.2], [[alpha].sub.2]}, {[[alpha].sub.1],[[alpha].sub.1] + [[alpha].sub.2],2[[alpha].sub.1] + [[alpha].sub.2],3[[alpha].sub.1] + 2[[alpha].sub.2],4[[alpha].sub.1] + 3[[alpha].sub.2],5[[alpha].sub.1] + 3[[alpha].sub.2], 5[[alpha].sub.1] + 4[[alpha].sub.2], [[alpha].sub.2]}, {[[alpha].sub.1],[[alpha].sub.1] + [[alpha].sub.2],2[[alpha].sub.1] + [[alpha].sub.2],3[[alpha].sub.1] + [[alpha].sub.2],4[[alpha].sub.1] + [[alpha].sub.2],5[[alpha].sub.1] + [[alpha].sub.2],5[[alpha].sub.1] + 2[[alpha].sub.2],[[alpha].sub.2]}.

The Cartan roots are, respectively, 2[[alpha].sub.1] + [[alpha].sub.2], [[alpha].sub.2]; [[alpha].sub.1] + [[alpha].sub.2],5[[alpha].sub.1] + 3[[alpha].sub.2]; [[alpha].sub.1],5[[alpha].sub.1] + 2[[alpha].sub.2]. Hence, in all cases, [Z.sub.q] [equivalent] [u(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

Row 13. Let [zeta] [member of] [G'.sub.5]. The braidings in this row are associated to the Lie superalgebra brj(2;5) [A5, [section]5.2]. If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] In this case the Cartan roots are [[alpha].sub.1], [[alpha].sub.1] + [[alpha].sub.2], 2[[alpha].sub.1] + [[alpha].sub.2] and 3[[alpha].sub.1] + [[alpha].sub.2], with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In [B.sub.q],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence the coproducts of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for some [a.sub.i] [member of] k. Thus, the following relations hold in [Z.sub.q]

[[[xi].sub.12],[[xi].sub.1]] = [a.sub.3] [[xi].sub.112,12]; [[[xi].sub.112,12],[[xi].sub.1]] = [a.sub.2] [[xi].sub.112]; [[[xi].sub.1],[[xi].sub.112,12]] = [[[xi].sub.12],[[xi].sub.112]] = 0.

Since

[a.sub.1] = - (1 - [[[zeta].sup.3]).sup.5] (1 + [[zeta]).sup.5] (1 + 62[zeta] - 15[[zeta].sup.2] - 87[[zeta].sup.3] + 70[[zeta].sup.4]) = 0; [a.sub.3] = - (1 - [[[zeta].sup.3]).sup.5][(1 + [zeta]).sup.8](4 - 8[zeta] - 19[[zeta].sup.2] - 3[[zeta].sup.3] - 50[[zeta].sup.4]) = 0,

the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not primitive, so [[xi].sub.1], [[xi].sub.12] generate [Z.sub.q]. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [N.sub.[alpha]1] = [N.sub.[alpha]1+[alpha]2] = 10, [N.sub.3[alpha]1+[alpha]2] = [N.sub.[alpha]1+[alpha]2] = 5. In [B.sub.q]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence the coproducts of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the generators of [Z.sub.q] are [[xi].sub.1] and [[xi].sub.12] and they satisfy the following relations

[[[xi].sub.12], [[xi].sub.1]] = [(1 + [zeta]).sup.10] (1 - [[[zeta].sup.3]).sup.5] [[xi].sub.1112], [[[xi].sub.1112],[[xi].sub.12]] =- [(1 + [zeta]).sup.5] (1 - [[[zeta].sup.3]).sup.5] [[xi].sub.112], [[[xi].sub.1],[[xi].sub.1112]] = [[[xi].sub.12],[[xi].sub.112]] = 0.

Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Row 14. Let [zeta] [member of] [G'.sub.20]. This row corresponds to type ufo(10). If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The Cartan roots are [[alpha].sub.1] and 3[[alpha].sub.1] + 2[[alpha].sub.2] with [N.sub.[alpha]1] = [N3.sub.[alpha]1+2[alpha]2] = 20. The elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are primitive; thus [[[xi].sub.12], [[xi].sub.112,12]] = 0 in [Z.sub.q] and [Z.sub.q] [equivalent] [U(([A.sub.1] [direct sum] [A.sub.1]).sup.+]). The same holds when the diagram of q is another one in this row: [Z.sub.q] [equivalent] [U(([A.sub.1] [direct sum] [A.sub.1]).sup.+]).

Row 15. Let [zeta] [member of] [G'.sub.15]. This row corresponds to type ufo(11). If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The Cartan roots are [[alpha].sub.1] and 3[[alpha].sub.1] + 2[[alpha].sub.2] with [N.sub.[alpha]1] = [N3.sub.[alpha]1]+]2[alpha]2] = 30. In Zq we have [[[xi].sub.12], [[xi].sub.112,12]] = 0, thus [Z.sub.q] [equivalent] [U(([A.sub.1] [direct sum] [A.sub.1]).sup.+]). The same result holds if we consider the other diagrams of this row.

Row 16. Let [zeta] [member of] [G'.sub.7]. This row corresponds to type ufo (12). If q has diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Also, [D.sub.q] = {[[alpha].sub.1],4[[alpha].sub.1] + [[alpha].sub.2],3[[alpha].sub.1] + [[alpha].sub.2],5[[alpha].sub.1] + 2[[alpha].sub.2],2[[alpha].sub.1] + [[alpha].sub.2],[[alpha].sub.1] + [[alpha].sub.2]} with [N.sub.[beta]] = 14 for all [beta] [member of] [D.sub.q]. In [B.sub.q] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

with [a.sub.i] [member of] k. For instance,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

because [zeta] [member of] [G'.sub.7]. Also,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not primitive elements in [B.sub.q]. Thus, [[xi].sub.1] and [[xi].sub.12] generates [Z.sub.q].

Also, in [Z.sub.q] we have

[[[xi].sub.12], [[xi].sub.1]] = [a.sub.1] [[xi].sub.112]; [[[xi].sub.1], [[xi].sub.112]] = [a.sub.2] [[xi].sub.1112]; [[[xi].sub.1],[[xi].sub.1112]] = [a.sub.4] [[xi].sub.11112]; [[[xi].sub.1],[[xi].sub.11112]] = [[[xi].sub.12],[[xi].sub.112]] = 0.

So, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the case of the diagram [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is generated by [[xi].sub.1], [[xi].sub.12] and

[[[xi].sub.12],[[xi].sub.1]] = [b.sub.1] [[xi].sub.112]; [[[xi].sub.12],[[xi].sub.112]] = [b.sub.2] [[xi].sub.112,12]; [[[xi].sub.12], [[xi].sub.112,12]] = [b.sub.3] [[xi].sub.(112,12),12]; [[[xi].sub.1],[[xi].sub.112]] = [[[xi].sub.12],[[xi].sub.(112,12),12]] = 0,

where [b.sub.1], [b.sub.2], [b.sub.3] [member of] [k.sup.x]. Hence, we also have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 4.1. The results of this paper are part of the thesis of one of the authors [RB], where missing details of the computations can be found.

References

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[AA] Andruskiewitsch, N; Angiono, I., Generalized root systems, contragredient Lie superalgebras and Nichols algebras, in preparation.

[AAR] Andruskiewitsch, N, Angiono, I., Rossi Bertone, F. The divided powers algebra of a finite-dimensional Nichols algebra of diagonal type, Math. Res. Lett., to appear.

[AAY] N. Andruskiewitsch, I. Angiono, H. Yamane. On pointed Hopf superalgebras, Contemp. Math. 544 (2011), 123-140.

[AN] N. Andruskiewitsch, S. Natale. Braided Hopf algebras arising from matched pairs of groups, J. Pure Appl. Alg. 182 (2003), 119-149.

[AS] N. Andruskiewitsch, H.-J. Schneider. Pointed Hopf algebras, New directions in Hopf algebras, MSRI series, Cambridge Univ. Press; 1-68 (2002).

[A1] I. Angiono. On Nichols algebras with standard braiding. Algebra and Number Theory, Vol. 3 (2009), 35-106.

[A2] I. Angiono. A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems. J. Eur. Math. Soc. 17 (2015), 2643-2671.

[A3] ---- On Nichols algebras of diagonal type. J. Reine Angew. Math. 683 (2013), 189-251.

[A4] ---- Nichols algebras of unidentified diagonal type, Comm. Alg 41 (2013), 4667-4693.

[A5] ---- Distinguished pre-Nichols algebras, Transform. Groups 21 (2016), 1-33.

[AY] I. Angiono, H. Yamane. The R-matrix of quantum doubles of Nichols algebras of diagonal type. J. Math. Phys. 56, 021702 (2015) 1-19.

[BD] Y Bespalov, B. Drabant. Cross Product Bialgebras Part II, J. Algebra 240 (2001), 445-504.

[CH] M. Cuntz, I. Heckenberger. Weyl groupoids with at most three objects. J. Pure Appl. Algebra 213 (2009), 1112-1128.

[GG] J. Guccione, J. Guccione. Theory of braided Hopf crossed products, J. Algebra 261 (2003), 54-101.

[H1] I. Heckenberger. The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164 (2006), 175-188.

[H2] ---- Classification of arithmetic root systems. Adv. Math. 220 (2009), 59-124.

[H3] ---- Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type. J. Alg. 323 (2010), 2130-2180.

[K] V. Kharchenko, A quantum analogue of the Poincare-Birkhoff-Witt theorem. Algebra and Logic 38 (1999), 259-276.

[L] G. Lusztig. Quantum groups at roots of 1. Geom. Dedicata 35 (1990), 89-113.

[RB] F. Rossi Bertone. Algebras cuanticas de potencias divididas. Thesis doctoral FaMAF, Universidad Nacional de Cordoba, available at www.famaf.unc.edu.ar/~rossib/.

[R] M. Rosso. Quantum groups and quantum shuffles. Inv Math. 133 (1998), 399-416.

Nicolas Andruskiewitsch

Ivan Angiono

Fiorela Rossi Bertone

(*) The work was partially supported by CONICET, Secyt (UNC), the MathAmSud project GR2HOPF

Received by the editors in March 2016.

Communicated by Y. Zhang.

2010 Mathematics Subject Classification : 17B37,16T20.

FaMAF-CIEM (CONICET),

Universidad Nacional de Cordoba,

Medina Allende s/n, Ciudad Universitaria,

5000 Cordoba, Republica Argentina.

email: (andrus--angiono--rossib)@mate.uncor.edu
Table 1: Lie algebras arising from Dynkin diagrams of rank 2.

Row              Generalized Dynkin diagrams

 1   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 2   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 3   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 4   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 5   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 6   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 7   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 8   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
 9   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
10   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
11   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
12   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
13   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
14   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
15   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
16   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Row              parameters               Type of [B.sub.q]

 1   q [not equal to] 1                   Cartan A
 2   q [not equal to] [+ or -] 1          Super A
 3   q [not equal to] [+ or -] 1          Cartan B
 4   q [??] [G.sub.4]                     Super B
 5   [zeta] [member of] [G.sub.3] [??] q  br(2, a)
 6   [zeta] [member of] [G'.sub.3]        Standard B
 7   [zeta] [member of] [G'.sub.12]       ufo(7)
 8   [zeta] [member of] [G'.sub.12]       ufo(8)
 9   [zeta] [member of] [G'.sub.9]        brj(2; 3)
10   q [??] [G.sub.2] [union] [G.sub.3]   Cartan [G.sub.2]
11   [zeta] [member of] [G'.sub.8]        Standard [G.sub.2]
12   [zeta] [member of] [G'.sub.24]       ufo(9)
13   [zeta] [member of] [G'.sub.5]        brj(2; 5)
14   [zeta] [member of] [G'.sub.20]       ufo(10)
15   [zeta] [member of] [G'.sub.15]       ufo(11)
16   [zeta] [member of] [G'.sub.7]        ufo(12)

Row  [n.sub.q] [congruent to] [g.sup.+]

 1   [A.sub.2]
 2   [A.sub.1]
 3   [B.sub.2]
 4   [A.sub.1] [direct sum] [A.sub.1]
 5   [A.sub.1] [direct sum] [A.sub.1]
 6   0
 7   0
 8   [A.sub.1]
 9   [A.sub.1] [direct sum] [A.sub.1]
10   [G.sub.2]
11   [A.sub.1] [direct sum] [A.sub.1]
12   [A.sub.1] [direct sum] [A.sub.1]
13   [B.sub.2]
14   [A.sub.1] [direct sum] [A.sub.1]
15   [A.sub.1] [direct sum] [A.sub.1]
16   [G.sub.2]
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Author:Andruskiewitsch, Nicolas; Angiono, Ivan; Bertone, Fiorela Rossi
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
Article Type:Report
Date:Jan 1, 2017
Words:8193
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